Application of Cartan Magic Formula (solving for vector field)

Question

I have been going through 'Lectures on Symplectic geometry' by AC da Silva and in the proof of the Moser Theorem (notable for the 'Moser Trick'), there is an application of the Cartan Magic Formula that eventually leads to the equation $$i_{v_t}\omega_t + \mu=0$$ where $\mu$ is a 1-form and it says we can 'solve for the vector field $v_t$' but I'm not sure how to proceed. Any insight would be much appreciated. It would also be great to know the reason for requiring that the 2-forms $\omega_0$ and $\omega_1$ be in the same cohomology class. I see it being used to say that $\omega_1-\omega_0 = d\mu$ for some 1-form $\mu$ but some intuition would be most welcome.

Answer 1

Let $(M,\omega)$ be a symplectic manifold. Then define $\tilde{\omega} \colon \mathfrak{X}(M) \to \Omega^1(M)$ by $\tilde{\omega}(X)(Y) := \omega(X,Y)$, called the tangent-cotangent bundle isomorphism. One can show that this is a smooth isomorphism (a bundle isomorphism if you see it as a map $TM \to T^*M$). Moreover, for any one form $\eta \in \Omega^1(M)$ we can solve $i_X\omega = \eta$ by setting $X := \tilde{\omega}^{-1}(\eta)$. Indeed, we compute

$$(i_X\omega)(Y) =\omega(X,Y) = \tilde{\omega}(X)(Y) = \eta(Y)$$

for every $Y \in \mathfrak{X}(M)$. For a proof of the tangent-cotangent bundle isomorphism, see theorem 2.40 in my notes on classical mechanics.

To answer your second question, in order to apply the moser trick you need the family $\omega_t$ to satisfy $$\frac{d}{dt} \omega_t = d\eta_t$$ for a smooth family of one forms $\eta_t$ (see for example Symplectic Topology by Dusa McDuff and Dietmar Salamon). Coinciding cohomology classes for example is a sufficient condition if you consider a linear interpolation.